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Theorem carden2b 9908
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9907 are meant to replace carden 10492 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΄) = (cardβ€˜π΅))

Proof of Theorem carden2b
StepHypRef Expression
1 cardne 9906 . . . . 5 ((cardβ€˜π΅) ∈ (cardβ€˜π΄) β†’ Β¬ (cardβ€˜π΅) β‰ˆ 𝐴)
2 ennum 9888 . . . . . . . 8 (𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ dom card ↔ 𝐡 ∈ dom card))
32biimpa 478 . . . . . . 7 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ 𝐡 ∈ dom card)
4 cardid2 9894 . . . . . . 7 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
53, 4syl 17 . . . . . 6 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
6 ensym 8946 . . . . . . 7 (𝐴 β‰ˆ 𝐡 β†’ 𝐡 β‰ˆ 𝐴)
76adantr 482 . . . . . 6 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ 𝐡 β‰ˆ 𝐴)
8 entr 8949 . . . . . 6 (((cardβ€˜π΅) β‰ˆ 𝐡 ∧ 𝐡 β‰ˆ 𝐴) β†’ (cardβ€˜π΅) β‰ˆ 𝐴)
95, 7, 8syl2anc 585 . . . . 5 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) β‰ˆ 𝐴)
101, 9nsyl3 138 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄))
11 cardon 9885 . . . . 5 (cardβ€˜π΄) ∈ On
12 cardon 9885 . . . . 5 (cardβ€˜π΅) ∈ On
13 ontri1 6352 . . . . 5 (((cardβ€˜π΄) ∈ On ∧ (cardβ€˜π΅) ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄)))
1411, 12, 13mp2an 691 . . . 4 ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄))
1510, 14sylibr 233 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) βŠ† (cardβ€˜π΅))
16 cardne 9906 . . . . 5 ((cardβ€˜π΄) ∈ (cardβ€˜π΅) β†’ Β¬ (cardβ€˜π΄) β‰ˆ 𝐡)
17 cardid2 9894 . . . . . 6 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
18 id 22 . . . . . 6 (𝐴 β‰ˆ 𝐡 β†’ 𝐴 β‰ˆ 𝐡)
19 entr 8949 . . . . . 6 (((cardβ€˜π΄) β‰ˆ 𝐴 ∧ 𝐴 β‰ˆ 𝐡) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
2017, 18, 19syl2anr 598 . . . . 5 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
2116, 20nsyl3 138 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅))
22 ontri1 6352 . . . . 5 (((cardβ€˜π΅) ∈ On ∧ (cardβ€˜π΄) ∈ On) β†’ ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅)))
2312, 11, 22mp2an 691 . . . 4 ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ Β¬ (cardβ€˜π΄) ∈ (cardβ€˜π΅))
2421, 23sylibr 233 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) βŠ† (cardβ€˜π΄))
2515, 24eqssd 3962 . 2 ((𝐴 β‰ˆ 𝐡 ∧ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
26 ndmfv 6878 . . . 4 (Β¬ 𝐴 ∈ dom card β†’ (cardβ€˜π΄) = βˆ…)
2726adantl 483 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = βˆ…)
282notbid 318 . . . . 5 (𝐴 β‰ˆ 𝐡 β†’ (Β¬ 𝐴 ∈ dom card ↔ Β¬ 𝐡 ∈ dom card))
2928biimpa 478 . . . 4 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ Β¬ 𝐡 ∈ dom card)
30 ndmfv 6878 . . . 4 (Β¬ 𝐡 ∈ dom card β†’ (cardβ€˜π΅) = βˆ…)
3129, 30syl 17 . . 3 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΅) = βˆ…)
3227, 31eqtr4d 2776 . 2 ((𝐴 β‰ˆ 𝐡 ∧ Β¬ 𝐴 ∈ dom card) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
3325, 32pm2.61dan 812 1 (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  βˆ…c0 4283   class class class wbr 5106  dom cdm 5634  Oncon0 6318  β€˜cfv 6497   β‰ˆ cen 8883  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8651  df-en 8887  df-card 9880
This theorem is referenced by:  card1  9909  carddom2  9918  cardennn  9924  cardsucinf  9925  pm54.43lem  9941  nnadju  10138  nnadjuALT  10139  ficardun  10141  ficardunOLD  10142  ackbij1lem5  10165  ackbij1lem8  10168  ackbij1lem9  10169  ackbij2lem2  10181  carden  10492  r1tskina  10723  cardfz  13881
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